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Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field

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  • Osada, Hirofumi

Abstract

We give a new sufficient condition of the quasi-Gibbs property. This result is a refinement of one given in a previous paper (Osada (in press) [18]), and will be used in a forthcoming paper to prove the quasi-Gibbs property of Airy random point fields (RPFs) and other RPFs appearing under soft-edge scaling. The quasi-Gibbs property of RPFs is one of the key ingredients to solve the associated infinite-dimensional stochastic differential equation (ISDE). Because of the divergence of the free potentials and the interactions of the finite particle approximation under soft-edge scaling, the result of the previous paper excludes the Airy RPFs, although Airy RPFs are the most significant RPFs appearing in random matrix theory. We will use the result of the present paper to solve the ISDE for which the unlabeled equilibrium state is the Airyβ RPF with β=1,2,4.

Suggested Citation

  • Osada, Hirofumi, 2013. "Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 813-838.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:813-838
    DOI: 10.1016/j.spa.2012.11.002
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    Cited by:

    1. Yosuke Kawamoto & Hirofumi Osada, 2019. "Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps," Journal of Theoretical Probability, Springer, vol. 32(2), pages 907-933, June.
    2. Honda, Ryuichi & Osada, Hirofumi, 2015. "Infinite-dimensional stochastic differential equations related to Bessel random point fields," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3801-3822.
    3. Katori, Makoto, 2014. "Determinantal martingales and noncolliding diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3724-3768.
    4. Osada, Hirofumi & Tanemura, Hideki, 2016. "Strong Markov property of determinantal processes with extended kernels," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 186-208.
    5. Yosuke Kawamoto & Hirofumi Osada, 2022. "Dynamical universality for random matrices," Partial Differential Equations and Applications, Springer, vol. 3(2), pages 1-51, April.

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